Question

If $A=(\begin{array}{cc}a& 1\\ 0& a\end{array})$, then for any n $\in$ N, $A^n$ equals

• A. $(\begin{array}{cc}na& n\\ 0& na\end{array})$
• B. $(\begin{array}{cc}{a}^n& n{a}^{n-1}\\ 0& a^n\end{array})$
• C. $(\begin{array}{cc}na& 1\\ 0& na\end{array})$
• D. $(\begin{array}{cc}{a}^n& n\\ 0& a^n\end{array})$

Answer 1

The correct option is B $(\begin{array}{cc}{a}^n& n{a}^{n-1}\\ 0& a^n\end{array})$

$A=\left(\begin{array}{cc}a& 1\\ 0& a\end{array}\right)$
$A^2=\left(\begin{array}{cc}a& 1\\ 0& a\end{array}\right)\left(\begin{array}{cc}a& 1\\ 0& a\end{array}\right)=\left(\begin{array}{cc}{a}^2& 2a\\ 0& a^2\end{array}\right)$
$A^3=A^2\cdot A=\left(\begin{array}{cc}a& 1\\ 0& a\end{array}\right)\left(\begin{array}{cc}{a}^2& 2a\\ 0& a^2\end{array}\right)=\left(\begin{array}{cc}{a}^3& 3a^2\\ 0& a^3\end{array}\right)$
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$\therefore A_n=\left(\begin{array}{cc}{a}^n& n{a}^{n-1}\\ 0& a^n\end{array}\right)$